PISCO: Self-Supervised k-Space Regularization for Improved Neural Implicit k-Space Representations of Dynamic MRI

TL;DR

This work tackles the challenge of overfitting in neural implicit k-space representations (NIK) for dynamic MRI when training data are limited. It introduces PISCO, a self-supervised k-space loss that enforces a global neighborhood consistency in k-space without requiring calibration data, integrated as $\mathcal{L}_{\mathrm{PISCO}}$ alongside the data-consistency loss in NIK training. The approach demonstrates improved spatiotemporal reconstructions across static and dynamic MRI datasets (upper leg, cardiac cine, abdomen), particularly at high accelerations ($R\geq 54$), often outperforming baseline NIK and, in some cases, rivaling or exceeding XD-GRASP in temporal fidelity while preserving spatial detail. The study also provides a thorough validation of PISCO's convergence, kernel design choices, and robustness, highlighting its potential as a versatile regularization tool for k-space–based MRI reconstruction and beyond.

Abstract

Neural implicit k-space representations (NIK) have shown promising results for dynamic magnetic resonance imaging (MRI) at high temporal resolutions. Yet, reducing acquisition time, and thereby available training data, results in severe performance drops due to overfitting. To address this, we introduce a novel self-supervised k-space loss function $\mathcal{L}_\mathrm{PISCO}$, applicable for regularization of NIK-based reconstructions. The proposed loss function is based on the concept of parallel imaging-inspired self-consistency (PISCO), enforcing a consistent global k-space neighborhood relationship without requiring additional data. Quantitative and qualitative evaluations on static and dynamic MR reconstructions show that integrating PISCO significantly improves NIK representations. Particularly for high acceleration factors (R$\geq$54), NIK with PISCO achieves superior spatio-temporal reconstruction quality compared to state-of-the-art methods. Furthermore, an extensive analysis of the loss assumptions and stability shows PISCO's potential as versatile self-supervised k-space loss function for further applications and architectures. Code is available at: https://github.com/compai-lab/2025-pisco-spieker

Figures (7)

• Figure 1: From GRAPPA Griswold_2002 to PISCO (proposed): GRAPPA (left) calibrates a weight vector $\mathrm{W_{ACS}}$ on a fully sampled autocalibration k-space (ACS, gray area, Eq. \ref{['eq:grappa_weightsolving']}), where gray lines indicate sampled k-space points. This vector determines the global neighborhood relationship $\mathrm{W}$ and is applied to derive target points in the undersampled k-space (US, white area). For PISCO (right), multiple random subsets of targets and patches ($\mathrm{T}_s$ and $\mathrm{P}_s$) are sampled to solve for $\mathrm{W}_s$ (Eq. \ref{['eq:PISCO_weightsolving']}). The parallel-imaging inspired self-consistency (PISCO) condition states that for an ideal k-space, all these weight vectors should equally converge to one single global neighborhood relationship $\mathrm{W}$ (Eq. \ref{['eq:PISCO_condition']}).
• Figure 2: Method overview (A/B/C corresponding to Sec. \ref{['sec:methods']}A/B/C). For simplicity the coil dimension is not visualized, but indicated by the matrix sizes. (A) PISCO: 1. From a k-space $y$, multiple subsets (one color = one subset) including pairs of target $y_i^\text{T}$ and patches $y_i^\text{P}$ are sampled. The patches consist of neighboring values, whereas any sampling shape can be assumed (see Sec. \ref{['sec:design_kernel']}). 2. Each subset is reshaped to a linear equation system and used to solve for the neighborhood relationship vector $\mathrm{W}_s$ using Eq. \ref{['eq:PISCO_weightsolving']} (Sec. \ref{['sec:design_weight']}). 3. The PISCO self-consistency of $y$ can be quantified using all subset weight vectors, e.g. either computing the distance Spieker_2024_pisco or the residual (proposed, visualized in B). (B) Residual-based PISCO as self-supervised k-space measure: For each subset $s$, $\mathrm{W}_s$ is used to estimate the targets $\Hat{\mathrm{T}}_s$ exclusively from their neighbors $\mathrm{P}_s$ and the residual to the actual targets $\mathrm{T}_s$ of $y$ is computed. The PISCO loss is determined as the weighted sum of all these residuals. (C) Integration of PISCO into NIK training: The "supervised" top represents the classical NIK-training Huang_2023, where actually acquired coordinates are sampled, predicted and compared to the acquisition signal using $\mathcal{L}_{DC}$. With PISCO, the perceptual field of NIK can be extended during training, because any patches (independent of acquisition trajectory) can be sampled and used for the self-supervised $\mathcal{L}_\text{PISCO}$ computation. In the present study, patches are sampled from a Cartesian grid, with alternating undersampling in the x- and y-dimension (only in y-dimension visualized).
• Figure 3: PISCO validation results. (A) Kernel Design: For each kernel geometry, multiple subsets of patches (one exemplary shown in blue) are sampled and solved for the weights (dark blue vector). The magnitude and phase of all weight vectors are stacked in the plot. To validate the PISCO condition, all weight vectors should result in the same solution, e.g. a vertical pattern is expected. Only the Cartesian kernel - shown in y-direction, but equal result in x-direction (90° turned) - fulfills this condition. (B) Weight Solving: Solution of subset weights (magnitude and phase) are stacked vertically (green and blue vectors are two examples). Top: Subsets consist of randomly sampled patches. Bottom: The sampled patches are sorted from the center to the outside of the k-space (white arrow) and then into subsets. This results in subsets with patches of similar k-space magnitude, leading to less noisy weight vector solutions overall. (C) Consistency Measure: Non-ideal k-spaces are simulated by adding increasing noise in image-space (left) and k-space (right). On the corresponding k-spaces, the distance-based Spieker_2024_pisco and residual-based (proposed) PISCO loss is computed and normalized by the maximum loss value. For image noise, both losses increase monotonically, while the residual-based loss is consistently sensitive. For k-space noise, only the residual-based loss monotonically increases, making it feasible for optimization towards the ideal k-space (where $\sigma$=0). (D) Validation of PISCO Fitting: K-Space (left) and image (middle) results for PISCO fitted undersampled k-spaces. Within k-space, inclusion of PISCO allows for derivation of missing k-space lines without any additional knowledge. A larger kernel (5x4, bottom) increases the perceptive field and fills more missing k-space lines. Reconstruction results lead to reduced undersampling using PISCO compared to the original undersampled image, visible in the images (middle) as well as difference images (right).
• Figure 4: Static upper leg: Qualitative and quantitative results for two exemplary acceleration factors (R10/20). NIK Huang_2023 results in increasing noise, particularly towards the center of the reconstruction. PISCO-dist Spieker_2024_pisco marginally improves reconstructions, but still remains noisy. With the proposed PISCO a better implicit representation is learned, that results in reduced noise and sharper structures (PSNR and FSIM $\uparrow\!$).
• Figure 5: Cardiac cine: Quantitative reconstruction results of 30 subjects for acceleration factors R15/R26/R52/R104 (with $\lambda$=0.01/0.05/0.1/0.15 for PISCO-dist/PISCO reconstruction, respectively). For R15, XD-GRASP25 offers least noisy reconstruction (PSNR) with high spatial and temporal resolution (FSIM-spat/FSIM-temp). For R26, NIK, PISCO-dist and PISCO lead to similar temporal results. At even larger acceleration factors, XD-GRASP25 performance decreases, as does NIK rapidly. PISCO-dist outperforms NIK, but does not reach XD-GRASP25. Yet, inclusion of the proposed PISCO significantly improves temporal structure compared to all other methods (R$>$26). At the same time, spatial reconstruction quality is maintained (R52) or even significantly improved (R104) compared to XD-GRASP25. All comparisons, except those marked with "N", are statistically significant (Wilcoxon signed rank test with False Discovery Rate correction at p$<$0.05).